FeatureLesson

GeometryGeometry

LessonMain

Find the coordinates of the image point for each given point andreflection line.

1. R(4, –5) across x = –2

2. S(–11, 2) across y = 1

3. T(0, 5) across x-axis

Lesson 9-2

ReflectionsReflections

XYZ has vertices X(–2, 3), Y(1, 1), and Z(2, 4). Draw XYZ and its reflection image in each line.

4. the x-axis 5. the line x = 5

R'(–8, –5)

S'(–11, 0)

T'(0, 5)

Lesson Quiz

9-3

FeatureLesson

GeometryGeometry

LessonMain

Lesson 9-3

(For help, go to the Skills Handbook, page 746 and Lesson 1-7.)

Use a protractor to draw an angle with the given measure.

1. 120 2. 90 3. 72

4. 60 5. 45 6. 36

7. Draw a segment, AB. Then construct A'B' congruent to AB.

RotationsRotations

Check Skills You’ll Need

Check Skills You’ll Need

9-3

FeatureLesson

GeometryGeometry

LessonMain

Solutions

Lesson 9-3

RotationsRotations

For problems 1–7, check students’ work.

Check Skills You’ll Need

9-3

FeatureLesson

GeometryGeometry

LessonMain

Lesson 9-3

RotationsRotations

9-3

To describe a rotation, you need to know the center of rotation (a point), the angle of rotation (a positive number of degrees), and whether the rotation is clockwise or counterclockwise. Unless stated otherwise, rotations in this book are counterclockwise.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 9-3

RotationsRotations

9-3

You can use the following two rules torotate a figure through x° about a point R:

• The image of R is itself (that is, R’ = R).

• For any point V, RV’ = RV and mVRV’ = x.

FeatureLesson

GeometryGeometry

LessonMain

Lesson 9-3

RotationsRotations

9-3

The center of a regular polygon is the point equidistant from its vertices. Segments that connect the center to the vertices divide the polygon into congruent triangles. You can use this fact to find rotation images of regular polygons.

FeatureLesson

GeometryGeometry

LessonMain

Copy LOB, and draw its image under a 60° rotation about C.

Step 1: Use a protractor to draw a 60° angle at vertex C with one side CO.

Lesson 9-3

RotationsRotations

Additional Examples

Drawing a Rotation Image

9-3

FeatureLesson

GeometryGeometry

LessonMain

Lesson 9-3

RotationsRotations

Step 3: Locate L and B in a similar manner. Then draw L O B .

(continued)

Quick Check

Additional Examples

Step 2: Use a compass to construct CO CO.

9-3

FeatureLesson

GeometryGeometry

LessonMain

Regular hexagon ABCDEF is divided into six equilateral triangles.

a. Name the image of B for a 240° rotation about M.

b. Name the image of M for a 60° rotation about F.

a. Because 360° ÷ 6 = 60°, each central angle of ABCDEF measures 60. A 240° counterclockwise rotation about center M moves point B across four triangles. The image of point B is point D.

b. AMF is equilateral, so AFM has measure 180 ÷ 3 = 60. A 60° rotation of AMF about point F would superimpose FM on FA, so the image of M under a 60° rotation about point F is point A.

Lesson 9-3

RotationsRotations

Quick Check

Additional Examples

Identifying a Rotation Image

9-3

FeatureLesson

GeometryGeometry

LessonMain

A regular 12-sided polygon can be formed by stacking

congruent square sheets of paper rotated about the same center on

top of each other. Find the angle of rotation about M that maps W to B.

Consecutive vertices of the three squares form the outline of a regular 12-sided polygon.

360 ÷ 12 = 30, so each vertex of the polygon is a 30° rotation about point M.

You must rotate counterclockwise through 7 vertices to map point W to point B, so the angle of rotation is 7 • 30°, or 210°.

Lesson 9-3

RotationsRotations

Quick Check

Additional Examples

Finding the Angle of Rotation

9-3

FeatureLesson

GeometryGeometry

LessonMain

Describe the image of quadrilateral XYZW for a composition

of a 145° rotation and then a 215° rotation, both about point X.

The two rotations of 145° and 215° about the same point is a total rotation of 145° + 215°, or 360°.

Because this forms a complete rotation about point X, the image is the preimage XYZW.

Lesson 9-3

RotationsRotations

Quick Check

Additional Examples

Compositions of Rotations

9-3

FeatureLesson

GeometryGeometry

LessonMain

Copy RST and point C. Draw the image for the given transformation about point C. Label the vertices of the image.

1. 75° rotation 2. composition of a 30° rotation and then a 150° rotation

Lesson 9-3

RotationsRotations

ABCDEFGH is a regular octagon. Name the image for the given rotation.

3. 135° rotation of A about O

G

F FG

5. 135° rotation of B about O

4. 270° rotation of DE about O

Lesson Quiz

9-3